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> arguably an invention A blind person would definitely disagree, having “invented” things he keeps bumping into, one after another… See, knowledge is never in

> arguably an invention A blind person would definitely disagree, having “invented” things he keeps bumping into, one after another… See, knowledge is never invented, it’s based on discovery; and mathematics is a form of knowledge. > anything about three Well, the discussion wasn’t about any particular number or concept, but if you want to define 3, you should look no further than the special property of a three-legged stool, the smallest polygon, or, if you look close enough, the proton. The general notion of the number, then, may come from a comparison (trying to see if there is a one-to-one correspondence) between the legs of a stool that is safe to sit on and one you risk breaking your neck if you try. If that’s full of “nuance,” I don’t know what isn’t.

danbruc 3y ago
A blind person would definitely disagree, having “invented” things he keeps bumping into, one after another… See, knowledge is never invented, it’s based on discovery; and mathematics is a form of knowledge. I was not talking about inventing the things themselves but logically dividing them up. For the blind person it makes not difference whether he bumps into a tree because you decided to divide the forest into several individual trees or whether he bumps into the entire forest. But if you want to count things, then it of course makes a huge difference whether there is just the forest or whether there is a collection of trees. This subdivision of the universe - or the forest - into several individual object - or trees - is arguably invented, not the universe or the forest itself. Knowledge is a thing you have, your awareness of some fact. Mathematics is not that, it is some form of fact you can be aware of. You can have knowledge about mathematics but it is not knowledge itself. If I name my dog Beethoven, that establishes the fact that the name of my dog is Beethoven, that is a kind of invention. If I tell you about this, you gain knowledge about the name of my dog. No discovery involved, neither when establishing the fact nor when you learn about it. [...] look no further than the special property of a three-legged stool, the smallest polygon [...] Okay, I found a stool that does not wobble on uneven ground and a triangular rock. Nothing about them on their own is related to three. You can of course explain to me what a leg is or the corner of a triangle and then ask me to form the sets of legs and corners and point out to me that the sets have the same number of elements, but there is lot of stuff going on here. You made me find things related to three, and I never doubted that there are such things, the number of dimensions of space would be another good candidate. But you mostly gloss over the hard part of actually extracting the natural numbers in general or three in particular. It is of course trivial in everyday language as we learn about pairing and counting things relatively early in our life and humanity has made use of those ideas for a long time. Look, the number of legs equals the number of corners. And there are as many of them as I have horses and apples. But notice that those sentences are full of ideas and words related to numbers - number of legs, equals, as many. But just as with the horses and apples, nothing about the non-wobbly stool or the smallest polygon - actually you mean the polygon with the fewest number of sides and note that there the idea of numbers already sneaked in again - is intrinsically related to three. It is the set of legs and the set of sides that are related to three, it is the carnality of the sets that behaves like the naturals. The process of forming those sets does a lot of heavy lifting to get you towards finding numbers in nature. And I do not think you can just brush that under the rug, you will have to justify that forming sets and looking at their cardinality is not something that humans invented.
- Koshkin 3y ago
  > it makes not difference whether he bumps into a tree because you decided Exactly. It makes no difference for him that you want to see a proof of individual trees' objective existence - because he already knows this for a fact! That's what that pesky objective reality does, sometimes forcing knowledge about itself upon us, whether we like it or not, or whether we would prefer some other "proof." The proof is in the pudding, as they say. > Mathematics is not that. Sure it is. One who knows about numbers knows more about the objective reality than those who don't. One who knows about Lie groups knows even more. > you gain knowledge about the name of my dog. No discovery involved That's not quite true: I discover that you have a dog (as long as you did not "invent" it). > Nothing about them on their own is related to three. It does, if you look at it from the right angle. There's a different thing at play here. While the number of legs could be easily matched with the corresponding number of apples, by itself this correspondence does not necessarily make any particular number stand out (although in some cases, like, say, in the case of a non-wobbly stool, a triangle, or the number of eyes and hands, it would - simply because there are many pairs of eyes, etc.); what's also at play here is different ways to look at numbers, which includes seeing them not only as "cardinals" (which is what you are still limiting yourself to) but also as "ordinals": the number 1 "stands out" as the smallest ordinal (greater than "nothing"), the number 2 is what follows it, etc. It is all these aspects combined that form the true content of the notion of the number.
  - danbruc 3y ago
    It makes no difference for him that you want to see a proof of individual trees' objective existence - because he already knows this for a fact! You are missing my point here. The blind guys knows he bumped into something, so something exists. He could just say he bumped into a part of the universe, he is not forced to say he bumped into a forest or a tree. He could even consider himself part of the universe and say one part of the universe bumped into another part of the universe, just as one part of you bumps into another part of you when you clap your hands. The consequence of that is that there are no distinct objects to count, it is just one really complex object, the universe, interacting with itself. Sure it is. One who knows about numbers knows more about the objective reality than those who don't. One who knows about Lie groups knows even more. No, that is a kind of map territory thing. You can have knowledge of mathematics but mathematics is not knowledge. You can have knowledge of my dos's name but my dog's name is not knowledge. It does, if you look at it from the right angle. Switching from the cardinals to the ordinals will not really make a difference, you are glossing over a lot of heavy lifting. I have to repeat myself, the non-wobbly stool is not related to three, it is the set of its legs that is related to three. On to get there, you have to single out parts of the stool and combine the parts into a set and the take about the cardinality of it. There are a lot of steps and concepts on that way which makes it at least very non-obvious how the number three was always there and is not just the result of that process. I am open to an example how I would find the ordinals in nature, I am not sure it will be any easier than with the cardinals. Non of the trees in the forest is the first one, you will have to impose an order an them. Maybe something with time, sunrises or days, they are at least already ordered.

danbruc 3y ago

A blind person would definitely disagree, having “invented” things he keeps bumping into, one after another… See, knowledge is never invented, it’s based on discovery; and mathematics is a form of knowledge. I was not talking about inventing the things themselves but logically dividing them up. For the blind person it makes not difference whether he bumps into a tree because you decided to divide the forest into several individual trees or whether he bumps into the entire forest. But if you want to count things, then it of course makes a huge difference whether there is just the forest or whether there is a collection of trees. This subdivision of the universe - or the forest - into several individual object - or trees - is arguably invented, not the universe or the forest itself. Knowledge is a thing you have, your awareness of some fact. Mathematics is not that, it is some form of fact you can be aware of. You can have knowledge about mathematics but it is not knowledge itself. If I name my dog Beethoven, that establishes the fact that the name of my dog is Beethoven, that is a kind of invention. If I tell you about this, you gain knowledge about the name of my dog. No discovery involved, neither when establishing the fact nor when you learn about it. [...] look no further than the special property of a three-legged stool, the smallest polygon [...] Okay, I found a stool that does not wobble on uneven ground and a triangular rock. Nothing about them on their own is related to three. You can of course explain to me what a leg is or the corner of a triangle and then ask me to form the sets of legs and corners and point out to me that the sets have the same number of elements, but there is lot of stuff going on here. You made me find things related to three, and I never doubted that there are such things, the number of dimensions of space would be another good candidate. But you mostly gloss over the hard part of actually extracting the natural numbers in general or three in particular. It is of course trivial in everyday language as we learn about pairing and counting things relatively early in our life and humanity has made use of those ideas for a long time. Look, the number of legs equals the number of corners. And there are as many of them as I have horses and apples. But notice that those sentences are full of ideas and words related to numbers - number of legs, equals, as many. But just as with the horses and apples, nothing about the non-wobbly stool or the smallest polygon - actually you mean the polygon with the fewest number of sides and note that there the idea of numbers already sneaked in again - is intrinsically related to three. It is the set of legs and the set of sides that are related to three, it is the carnality of the sets that behaves like the naturals. The process of forming those sets does a lot of heavy lifting to get you towards finding numbers in nature. And I do not think you can just brush that under the rug, you will have to justify that forming sets and looking at their cardinality is not something that humans invented.

Koshkin 3y ago
> it makes not difference whether he bumps into a tree because you decided Exactly. It makes no difference for him that you want to see a proof of individual trees' objective existence - because he already knows this for a fact! That's what that pesky objective reality does, sometimes forcing knowledge about itself upon us, whether we like it or not, or whether we would prefer some other "proof." The proof is in the pudding, as they say. > Mathematics is not that. Sure it is. One who knows about numbers knows more about the objective reality than those who don't. One who knows about Lie groups knows even more. > you gain knowledge about the name of my dog. No discovery involved That's not quite true: I discover that you have a dog (as long as you did not "invent" it). > Nothing about them on their own is related to three. It does, if you look at it from the right angle. There's a different thing at play here. While the number of legs could be easily matched with the corresponding number of apples, by itself this correspondence does not necessarily make any particular number stand out (although in some cases, like, say, in the case of a non-wobbly stool, a triangle, or the number of eyes and hands, it would - simply because there are many pairs of eyes, etc.); what's also at play here is different ways to look at numbers, which includes seeing them not only as "cardinals" (which is what you are still limiting yourself to) but also as "ordinals": the number 1 "stands out" as the smallest ordinal (greater than "nothing"), the number 2 is what follows it, etc. It is all these aspects combined that form the true content of the notion of the number.
- danbruc 3y ago
  It makes no difference for him that you want to see a proof of individual trees' objective existence - because he already knows this for a fact! You are missing my point here. The blind guys knows he bumped into something, so something exists. He could just say he bumped into a part of the universe, he is not forced to say he bumped into a forest or a tree. He could even consider himself part of the universe and say one part of the universe bumped into another part of the universe, just as one part of you bumps into another part of you when you clap your hands. The consequence of that is that there are no distinct objects to count, it is just one really complex object, the universe, interacting with itself. Sure it is. One who knows about numbers knows more about the objective reality than those who don't. One who knows about Lie groups knows even more. No, that is a kind of map territory thing. You can have knowledge of mathematics but mathematics is not knowledge. You can have knowledge of my dos's name but my dog's name is not knowledge. It does, if you look at it from the right angle. Switching from the cardinals to the ordinals will not really make a difference, you are glossing over a lot of heavy lifting. I have to repeat myself, the non-wobbly stool is not related to three, it is the set of its legs that is related to three. On to get there, you have to single out parts of the stool and combine the parts into a set and the take about the cardinality of it. There are a lot of steps and concepts on that way which makes it at least very non-obvious how the number three was always there and is not just the result of that process. I am open to an example how I would find the ordinals in nature, I am not sure it will be any easier than with the cardinals. Non of the trees in the forest is the first one, you will have to impose an order an them. Maybe something with time, sunrises or days, they are at least already ordered.